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"a problem for which" is correct and usable in written English.
You can use it to introduce a phrase that gives the solution to a problem. For example: "A problem for which many people are searching for the solution is poverty."
Exact(58)
It is a problem for which Cumberbatch's performance is an almost complete solution.
Others may stimulate appetite (a problem for which America has no shortage of fixes: see article).
Debugging optimized code is a problem for which a widely accepted solution has yet to be found.
Here, at last, is a problem for which the slogan "Mend it, don't end it" really does make sense.
One example concerns the effort to reduce domestic violence, a problem for which there is no quick fix.
But, living in a competitive and utilitarian society, he doesn't have sufficient talent to be treated as an individual, a problem for which there is no solution.
But a school district had to show that it had a problem for which the particular drug testing program was the solution, the appeals court said.
If these hominids are indeed a separate species they remind us that life is a problem for which the solutions are many.
Across Europe, criminal justice officials acknowledge that they are seeking to engineer solutions to a problem for which there is scant scientific data, and no guarantees of success.
Similar(2)
"It's a medical problem — for which a treatment exists, but for which at the moment a cure does not," he said.
In theory this is not a problem for ICE, which supposedly shares Mr. Ammiano's priorities.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com